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THE DFT SINGLE-BIN FREQUENCY RESPONSE TO A REAL COSINE INPUT

Now that we understand the DFT's overall N-point (or N-bin) frequency response to a real cosine of k cycles per sample interval, we conclude this chapter by determining the frequency response of a single DFT bin. We can think of a single DFT bin as a kind of bandpass filter, and this useful notion is used, for example, to describe DFT scalloping loss (Section 3.10), employed in the design of frequency-domain filter banks, and applied in a telephone frequency multiplexing technique known as transmultiplexing[15]. To determine a DFT single-bin's frequency response, consider applying a real xr(n) cosine sequence to a DFT and monitoring the output magnitude of just the m = k bin. Let's say the initial frequency of the input cosine sequence starts at a frequency of k<m cycles and increases up to a frequency of k>m cycles in the sample interval. If we measure the DFT's m = k bin during that frequency sweep, we'll see that its output magnitude must track the input cosine sequence's continuous spectrum, shown as the shaded curves in Figure 3-45.

Figure 3-45. Determining the output magnitude of the mth bin of an N-point DFT: (a) when the real xr(n) has k = m–2.5 cycles in the time sequence; (b) when the real xr(n) has k = m–1.5 cycles in the time sequence; (c) when the real xr(n) has k = m cycles in the time sequence; (d) the DFT single-bin frequency magnitude response of the m = k bin.

 

 

 

Table 3-4. Various Forms of the Positive Frequency Amplitude Response of the DFT to a Real Cosine Input Having k Cycles in the Sample Interval

Description

Expression

Real input DFT amplitude response in terms of the integral frequency variable m [From Eq. (3-73)]



 

Alternate form of the real input DFT amplitude response in terms of the integral frequency variable m [based on Eq. (3-49)]

Equation 3-75



 

Amplitude normalized real input DFT response in terms of the integral frequency variable m

Equation 3-76



 

Real input DFT response in terms of the sample rate fs in Hz

Equation 3-77



 

Amplitude normalized real input DFT response in terms of the sample rate fs in Hz

Equation 3-78



 

Amplitude normalized real input DFT response in terms of the sample rate ws in radians/s

Equation 3-79



 

Figure 3-45(a) shows the m = k bin's output when the input xr(n)'s frequency is k = m–2.5 cycles per sample interval. Increasing xr(n)'s frequency to k = m–1.5 cycles per sample interval results in the m = k bin's output, shown in Figure 3-45(b). Continuing to sweep xr(n)'s frequency higher, Figure 3-45(c) shows the m = k bin's output when the input frequency is k = m. Throughout our input frequency sweeping exercise, we can see that the m = k bin's output magnitude must trace out the cosine sequence's continuous spectrum, shown by the solid curve in Figure 3-45(d). This means that a DFT's single-bin frequency magnitude response, to a real input sinusoid, is that solid sinc function curve defined by Eqs. (3-74) through (3-79).

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Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183
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